34 SOIL CONDITIONS AND PLANT GROWTH 



Germany. Pfeiffer ^ insists that the curves most be profoundly 

 modified by other limiting factors, while Frohlich takes excep- 

 tion to the method of calculation.^ Mitscherlich apparently 

 recognises the force of these criticisms, for he now admits 

 that Liebig's Law of the Minimum is not a correct expression 

 of the facts : l:he yield is determined not by the one factor 

 which is lacking, but by all the factors. Baule (9) has deve- 

 loped an equation on these lines which, he claims, agrees 

 satisfactorily with the experimental results. 



Curves of similar type have been obtained for variations in 

 water supply and in temperature, but owing to greater experi- 

 mental difficulties the data are less reliable, and, therefore, not 

 susceptible to mathematical treatment. The temperature re- 

 sults are of interest, in that they bring out an important differ- 



^ Th. Pfeiffer, E. Blanck, and M. Fliigel, Wasser und Licht als Vegetations- 

 f actor en undihre Beziehungen zum Gesetze von Minimum (Landw. Versuchs-Stat., 

 1912, Ixxvi., 169-236. See also 224^). 



2 The method of calculation is as follows : Obtain two equations by sub- 

 stituting two of the numerical values ofx and y obtained experimentally. Calling 

 these numbers x^, x^, etc., the equations are 



log, {A -y^) =c - kx^ (i) 



loge {A - y^) = c - kxz (2) 



Then by subtraction log (A - y^) - log (A - ^2) — k (x^ - x-^) . . . (3) 



Obtain another equation like (3) but select the numerical values so that 



J*JJ "~ ^rt ^^ Xft ~~ X 



loge (A - y.,) - loge (A - y^) = k (x^ - x^) . . . (4) 



By subtracting (4) from (3) loge (A - y-^) + loge (A - y^) = 2 loge (A - y^, 



(A - J3) (A - y^) _ 

 '■'' (A-y,Y -' .... (5) 



Since ^i,^2» ^""^ Js ^"^^ ^^' known, the value of A is easily calculated. 

 The value of ^ is then found from equation (3) 



^ loge(A - y^) - loge(A - y^) 



X2 — X-y 



As all the quantities on the right-hand side are known the value of k is 

 readily obtained. A difficulty is that different values for A are, in fact, obtained 

 when other equations like (3) are worked out. If there were a very large number 

 of points, a probable value for A could be obtained : with a small number, such 

 as almost necessarily are obtained in practice, some selection apparently has to 

 be made, which is objectionable. This was done by Mitscherlich in the case 

 quoted in Table V. 



